simple? question about discrete to continuous translation

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

Hi all,
I think the answer to my question is probably very simple, but I got confused so decided to ask for some help.

I have the following equation:
$y = a \sum_{i=1}^N \frac{\alpha_i \phi_i}{\ln(\phi_i)}$
where $\alpha_i$ is the discrete probability of $\phi_i$ and of course $\sum_{i=1}^N \alpha_i = 1$

I would like to translate this to an equation where the probability density function is continuous instead of discrete. I think it should be something like this:

$y = a \int P(\phi) d\phi$

where $P(\phi)$ is my probability density function for $\phi$ but I am not sure and I don't know what function to use for $P(\phi)$ then. Can anybody help / point me in the right direction?!

Thanks!

tashirosgt · **Posted:** Sun, 1 Apr 2012 16:51:37 UTC

michielm wrote:

I would like to translate this to an equation where the probability density function is continuous instead of discrete

That doesn't define a precise goal. I think you need to explain the significance of the discrete equation.

You can certainly ask the purely mathematical question of "How do I find a continuous function f(x) that when integrated from x = 0 to x = N gives me the same numerical answer as summing a given discrete function g(x) from 1 to N?". However, the answer to this is generally not what one wants in an application of math to a real world problem. Usually, the question is "If I have a given function g(x) defined on x = 1,2,..N that approximates an answer to a problem, how do I define a continuous function f(x) so that the integral of f(x) from x = 0 to x = N give me the exact answer to the problem?"

Shadow · **Posted:** Sun, 1 Apr 2012 18:18:53 UTC

tashirosgt wrote:

michielm wrote:

I would like to translate this to an equation where the probability density function is continuous instead of discrete

That doesn't define a precise goal. I think you need to explain the significance of the discrete equation.

You can certainly ask the purely mathematical question of "How do I find a continuous function f(x) that when integrated from x = 0 to x = N gives me the same numerical answer as summing a given discrete function g(x) from 1 to N?". However, the answer to this is generally not what one wants in an application of math to a real world problem. Usually, the question is "If I have a given function g(x) defined on x = 1,2,..N that approximates an answer to a problem, how do I define a continuous function f(x) so that the integral of f(x) from x = 0 to x = N give me the exact answer to the problem?"

No, the problem is even more convoluted than that, he wants P to be a continuous density function, which--if $\phi$ is a finite random variable (and it *does* appear to be so as given)--is impossible.

mathematic · **Joined:** Fri, 1 Jul 2011 01:17:26 UTC **Posts:** 322

You could use Dirac delta functions. P would be a linear combination of the delta functions. For mathematical rigor, the distribution function would be a step function with jumps at the discrete values of the argument, and the size of the jumps would be the probabilites

Shadow · **Posted:** Sun, 1 Apr 2012 22:21:19 UTC

mathematic wrote:

You could use Dirac delta functions. P would be a linear combination of the delta functions. For mathematical rigor, the distribution function would be a step function with jumps at the discrete values of the argument, and the size of the jumps would be the probabilites

Yes, but he wants a continuous function, which cannot happen.

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

Let me try to explain what I would like to have:

The function I have now is discrete as you've all seen.
What I would like to have is a continuous probability density function that would show the same curve as the discrete function if I would take the discrete function with large N and distribute the output over bins. Much like the graphs shown on this page http://en.wikipedia.org/wiki/Central_limit_theorem.

I hope this explanation helps!

Shadow · **Posted:** Mon, 2 Apr 2012 14:31:03 UTC

michielm wrote:

Let me try to explain what I would like to have:

The function I have now is discrete as you've all seen.
What I would like to have is a continuous probability density function that would show the same curve as the discrete function if I would take the discrete function with large N and distribute the output over bins. Much like the graphs shown on this page http://en.wikipedia.org/wiki/Central_limit_theorem.

I hope this explanation helps!

You mean you want to know what the limiting distribution is (again what you have asked for is impossible, no matter how large N is you'll never get equality, only a better and better approximation.)

Without knowing the a priori distribution on the $\phi_i^{(N)}$ , there's no way to tell. Each time you increase

the weights on the new $\phi_i$ should change and without more information, it is impossible to tell what it will converge to, if that random variable sequence even DOES converge to another random variable (which is not guaranteed!)

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

Yes, I want the limiting distribution. Sorry for making myself clear in such a poor manner.

I now understand that I should know $\phi_i^{(N)}$ to be able to do this. I actually measure $\phi_i^{(N)}$ experimentally in this case and have tried for different large N that the distribution at least appears to be the same.

So let's assume that I know $\phi_i^{(N)}$ and I know that it converges for large N. How would I obtain the approximate continuous function? Does this involve curve fitting or is there a more appropriate and direct way of doing it?

Shadow · **Posted:** Mon, 2 Apr 2012 17:45:43 UTC

michielm wrote:

Yes, I want the limiting distribution. Sorry for making myself clear in such a poor manner.

I now understand that I should know $\phi_i^{(N)}$ to be able to do this. I actually measure $\phi_i^{(N)}$ experimentally in this case and have tried for different large N that the distribution at least appears to be the same.

So let's assume that I know $\phi_i^{(N)}$ and I know that it converges for large N. How would I obtain the approximate continuous function? Does this involve curve fitting or is there a more appropriate and direct way of doing it?

Generally, unless you know the distribution of the $\phi_i^{(N)}$ satisfy say the conditions of the central limit theorem, there's not much that can be said. You need to be in the situation where you can apply one of the big results on distributional convergence. In all probability, it will likely converge to the normal distribution, as this is usually the case.

fam · **Joined:** Tue, 16 Aug 2011 23:38:56 UTC **Posts:** 124

michielm wrote:

Hi all,
I think the answer to my question is probably very simple, but I got confused so decided to ask for some help.

I have the following equation:
$y = a \sum_{i=1}^N \frac{\alpha_i \phi_i}{\ln(\phi_i)}$
where $\alpha_i$ is the discrete probability of $\phi_i$ and of course $\sum_{i=1}^N \alpha_i = 1$

I would like to translate this to an equation where the probability density function is continuous instead of discrete. I think it should be something like this:

$y = a \int P(\phi) d\phi$

where $P(\phi)$ is my probability density function for $\phi$ but I am not sure and I don't know what function to use for $P(\phi)$ then. Can anybody help / point me in the right direction?!

Thanks!

Check the Weibull Distribution. The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions.

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

Thanks! I'll definitely try that

tashirosgt · **Posted:** Tue, 3 Apr 2012 03:26:40 UTC

michielm wrote:

Yes, I want the limiting distribution. Sorry for making myself clear in such a poor manner.

You need to provide more details in order to get good advice. How to interpret and use data depends a lot on how sampling is done.

It isn't clear from your description whether you are measuring a random variable that has finite limits or whether you are measuring something whose max or min increases as you increase the number of bins. For example, if you were measuring a voltage over a 10 second interval, it would presumably have some max and min value and when you use more and more "bins", the time between measurements is getting shorter and shorter. On the other hand, if you were counting the number of heads in N tosses of a coin and you use a bin size of 1 and increase the number of bins by increasing N, then the possible max number of heads increases.

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

What I am measuring is the following:
I have a surface with circles semi-randomly positioned (random placement but with the constraint that circles cannot overlap) and I tesselate that surface with voronoi tesselation such that I get a large number of polygon cells, all with their own area. The size of the circle divided by the area of the polygon (the area fraction) is what I want the distribution of. Because I don't count circles on the edge of my domain an increase in the amount N of circles (thus cells) will not increase the minimum and maximum because they remain between 0 (polygon is infinite) and 1 (polygon exactly follows the circle).

So I have a random variable with finite limits.

tashirosgt · **Posted:** Thu, 5 Apr 2012 22:30:05 UTC

michielm wrote:

I have a surface with circles semi-randomly positioned (random placement but with the constraint that circles cannot overlap) and I tesselate that surface with voronoi tesselation such that I get a large number of polygon cells, all with their own area.

It isn't clear what points you use to define the tesselation.

Quote:

The size of the circle divided by the area of the polygon (the area fraction) is what I want the distribution of.

What polygon are you talking about? Is there supposed to be a unique polygon associated with each cir

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

1) I use the centers of the circles as basis for my voronoi tesselation

2) Because I use the centers each circle will have 1 polygon that is drawn around it which I associate with it and use in the area fraction calculation

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simple? question about discrete to continuous translation

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